Araki, S. and Toda, H.
Osaka J. Math.
2 (1965), 71-115
MULTIPLICATIVE STRUCTURES IN MOD q
COHOMOLOGY THEORIES I.
SHORO
ARAKI and HIROSI TODA
(Received March 24, 1965)
Introduction. In the course of developments of algebraic topology,
one of the advantages to use the cohomology theory (in the ordinary
sense) rather than the homology theory was the existence of multiplicative structures, i. e., cup products. Cup products exist not only in
the integral cohomology but also in the cohomology with any coefficients
whenever the coefficient domain has a multiplicative structure. Recently
it is known that for any general cohomology theory one can associate
cohomology theories with coefficients. Since there are important general
cohomology theories with multiplicative structures such as ϋΓ-theories,
one can expect to introduce multiplicative structures in their associated
cohomology theories with coefficients. The present work is directed to
introduce and to study multiplicative structures in cohomology theories
with coefficients. However, since coefficients are limited to finitely
generated groups at the present time, the most important cases are those
with coefficients in some finite cyclic groups, i.e., Zqy q>l. Henceforth
our research is limited to mod q cohomology theories to avoid complexity of discussions.
To introduce multiplications in mod q cohomology theories it is
important to check a connection with the multiplication in the original
cohomology theory. Since there is the notion of "reduction mod q"
also in general cohomology theories, we postulate this connection as
the compatibility through reduction mod q, postulation (Λ^. In the
ordinary mod q cohomology theory, the mod q Bockstein homomorphism
works as a derivation. Since this property has been proved to be much
useful, we postulate a corresponding property also in general mod q
cohomology theories, postulation (Λ2). Proof of associativity of multiplications in mod q cohomology theories are very round about even if
it is possible. But, to get some uniqueness type theorems, it is sufficient
to postulate a weaker form of associativity, which we call "quasiassociativity," postulation (Λ3). We have an example of mod q
72
S. ARAKI and
H.
TODA
cohomology theory which have a multiplication satisfying (Λx), (Λ2) and
(Λ3) but no commutative one, i. e., complex iΓ-theory mod 2. So that
we postulate nothing about commutativity. A multiplication in a mod q
cohomology theory satisfying (Λ2), (Λ2) and (Λ3) is called "an admissible
multiplication."
If q^2 (mod 4), then there exist an admissible multiplication always
whenever the multiplication in the original cohomology theory is given
and associative (Theorem 5.9). But, if q=2 (mod 4), some condition
is necessary for the existence of admissible multiplications. A sufficient
condition for this is obtained (Theorem 5.9). In case q=2 (mod 4), a
necessary and sufficient condition for the existence of a multiplication
satisfying only (Λx) is obtained (Corollary 5.6). There are examples
which do not satisfy this condition, e. g., ϋCO-cohomology theory of real
vector bundles.
Not only the existence but also a uniqueness-type theorem of
admissible multiplications is discussed, which states that admissible
multiplications are in a one-to-one correspondence with elements of a
group which is specific to the considered cohomology theory under the
assumption that the original multiplication is commutative and associative
(Corollary 3.10). In case of ordinary cohomology this group consists
only of zero, hence true uniqueness holds. In case of complex ϋΓ-theory,
this group is Zqy hence there are q different admissible multiplications
in mod q if-theory. In case of the i£O-theory, this group consists only
of zero if q is odd, hence true uniqueness holds for odd q.
In § 1 we summarize some known basic properties of stable homotopy groups of two C TF-complexes, which is necessary as preliminaries.
In § 2 we exhibit some elements of additive mod q cohomology theories :
reduction mod q, Bockstein homomorphisms, universal coefficient
sequences, cohomology maps induced by coefficient homomorphisms, etc.
In § 3 we develop an axiomatic approach to multiplications in mod q
cohomology theories, and arrive to the notion of admissible multiplications. Uniqueness-type theorems and deviations from the commutativities are discussed. In § 4 we compute some stable homotopy groups
and make preparations for the existence theorem of admissible multiplications from the homotopy theoretical point of view. § 5 is devoted
to the proof of the existence theorem of admissible multiplications by
constructing a multiplication.
In a subsequent paper with the same title we will discuss associativities, commutativities and multiplications in Bockstein spectral
sequences, at least for the if-theory.
MULTIPLICATIVE STRUCTURES IN MOD q COHOMOLOGY THEORIES I.
1.
73
Preliminaries
1.1. First we shall fix some notations :
X/\ Y
the reduced join of two spaces X and Y with base
points,
f/\g
the reduced join of two base-point-preserving maps /
and g,
SX=XΛS
the (reduced) suspension of X>
1A (or simply 1):A^A
an identity map of A into itself,
T(A, B) (or simply T): A/\B->B/\A
a map switching factors,
n
n 1
n 1
1
n
S A = S(S - A) = AAS - AS = AAS
an n-ίold suspension of A,
Snf=fΛlsn
an n-ίold suspension of a map /,
q: S1-^S1
for an integer q to denote a map of degree q given
by q{t} = {qt} for {t mod 1} €ΞS\
1
Denote by {A, B} the stable homotopy groups of C ίF-complexes A
and B with base point, i. e., the limit of the sets of the base-pointpreserving homotopy classes of maps :SnA-^SnB with respect to suspensions, endowed with the usual structure of an abelian group. Obviously
we have
(1.1).
{A B} - {S"A9 SnB}}
n = 0,1,2,
.
For a map f:SnA^SnB>
n^O, we denote by the same letter / the
stable homotopy class represented by /, i. e., / e {^4, B} when there
arises no confusion. For example,
1 = lΛe
{A
and
A}
T=
Γ(A
B)ΪΞ{AΛB,
BΛA}
for the classes of the identity map and the switching map,
2
1
y(Ξ{S , S }
and
for the classes of the Hopf maps y:S*-^S2
1.2.
4
1
^{S ^ }
and i/:S 7 ->S\
The composition of αe{il,B} and β<^{B, C}, denoted by
/?oα
or simply by
m
n
W
W
n
n
is defined as the class of S goS f, where /:S A-^S £ and g:S B-+S C
are respectively representatives of a and /3. Tnis definition is independent of the choices of / and g.
For α ε { i l , 5 } and α ' e ^ , 5'}, their reduced join
74
S. ARAKI and H. TODA
is defined as follows: choosing representatives f:SmA->SmB
and
g:SnA'-^SnB'
of a and a' respectively, a Ace' is the class of the composition
ΆSm+H
=
This definition is also independent of the particular choices of /
and g.
We have obviously
(1.2). The composition and the reduced join are bilinear and the relations
<βa)/\(β'a?) =
and
(βAβ')(aAaf)
(a A a') T(A', A) = T(B', B)(a' A a)
hold for
« G {A β}, α 7 e {A\ β 7 }, /3G {B, C} and βf^{B\
C'} .
We regard that, for any α e { i , B} and the class l M e{S M , Sn} of the
identity map, the relation
Sna = aAln = a
(1.3)
holds via the identification (1.1). In general lH/\a differs from aAln>
but, if αG{S ί + s , Sp} and /3e{S*+/, S*}, then we see from (1.2) that
aAβ = (-iYp+s»aβ
(1. 4)
= {-lytβa
,
ns
lnAcc = (—l) a .
in particular
By the bilinearity of compositions, {A, A} (or 2 Λ {S"A, A}) forms
a ring (or a graded ring) with the composition as the multiplication.
Also the formula
/3*(α) = α*G8) = βa,
αG {A, B}, β(Ξ {B, C) ,
defines homomorphisms
/3* : {A, B} -> {Λ C}
1.3.
Let f:A->B
and
α* : {5, C} -> {A, C}.
be a map of finite CW-complexes and let
C r = B U / C4
be the mapping cone of /.
We have Puppe's exact sequence [7]
MULTIPLICATIVE STRUCTURES IN MOD q COHOMOLOGY THEORIES I.
(1. 5).
75
Sn+1f*
SV
SHi*
U {S»+1A, X}
> {S"Cf9 X}
>
sy*
n
{S B, X} - ^
{SnA, X} — ...
for any X, where π:Cf-*SA
is the map collapsing B to a point and
i:B->Cf
the canonical inclusion. As a dual of (1.5) we have the
following exact sequuence for finite dimensional CTF-complex Y
(1. 50
.-
Both sequences will be called the exact sequences associated with
the cofibration
1.4. Let Mq be a co-Moore space of type (Zq, 2), where q is an
integer > 1 . To fix a space Mq used in the sequel we put
Denote by
(1.6)
nq (or π):Mq->S2
and
iq (or O' S'-^A^
the map collapsing S1 to a point and the canonical inclusion.
The following theorem is basic in our later discussions.
q.
Theorem 1.1. // q^2 (mod 4), lM(={Mqy Mq}, M=Mq, is of order
If q=2 (mod 4), then qΛM=iqηπq
for the class of Hopf map
1
\ S }^Z2 and the order of 1M is 2q.
We sketch the proof (for details, see [ 3 ] or [9]).
In the exact
sequence
{S2, Mq} ^-+ {Mq, Mq} - ^ {S\ Mq)
i*(lM) = i is of order q, and qΛM is in the image of 7r*. i%: {S2, S1}
->{S2,Mq} is an epimorphism. Thus q-lM=0 or iηπ. If q is odd then
iyπ=0. Let q be even. q lM=iyπ if and only if the functional S#2
2
operation associated with qΛM is not zero, i.e., Sq 4=0 in MqΛMq
z
1
1
= SMq[Jq.1C(SMq).
By Cartan's formula, Sq = Sq ΛSq Φθ if and only
if q=2 (mod 4). This proves the theorem.
76
S. ARAKI and H. TODA
1.5. Apply (1.5) and (1.50 for the cofibration
SX-^lχ/\Mq
2
>S X, where XAMq=SX\JfC(SX)
and f=lxAq = qΛsx : SX-+SX
induces the q times of the identity map. Then we have the following
two exact sequences
2
(
(1. 7)
0 - {S X9 W} ®Zq - ^ Γ {XA Mq, W)
Tor({SX, W}9 Zq)->0,
(1.70
0->{Xf SX}®Zq
(Λ
A
Λ
Tor({F,
for finite dimensional F.
By Theo. 1.1 and (1.2) we see that the order of the identity class
IXΛIM of XAMq is a divisor of q if q^β2 (mod 4) and is a divisor of
2q in general. Hence
(1.8) {XAMqyW} and {Y, XAMq)
Zq-modules if q^2 (mod 4).
are Z2q-modules in general, and are
From this we see that the sequences (1.7) and (1.70 split for odd
prime q.
2. mod q cohomology theories
2.1. By a cohomology theory we understand, throughout the
present work, a general cohomology theory defined on the category of
pairs (X, A) of finite CW-complexes (or of the same homotopy type).
Each cohomology theory h has its reduced cohomology theory h defined
on the category of finite C ^-complexes with base points. The correspondence h->h is bijective, and the postulations for h have an
equivalent form of postulations for h [10] thereby the excision axiom
is replaced by the suspension isomorphism
for all i.
Put
Then A* is a functor of Z-graded abelian groups. (In case h = K or
KO we use ** instead of * to denote the Z-graded groups so as to avoid
confusions with the periodic cohomologies.) For the sake of simplicity
MULTIPLICATIVE STRUCTURES IN MOD q COHOMOLOGY THEORIES I.
77
A*(/) is denoted by
for any map / : Y—>X preserving base points. From the commutativity
S/*ocr=<7c>/* (the naturality of σ) follows that / * depends only on the
stable homotopy class of /.
Let A, B be finite C W-complexes (with base points). For any
element a of {A, B}, we define a homomorphism
a** : A* (XA B) -* h* (XA A)
by the formula
where f:SnA-+SnB
is a map representing α. The commutativity
σ(lΛ/)*==(lΛιS/)*<r shows that the definition does not depend on the
choice of /. We have easily
(2.1). i) α** is the identity homomorphism if a is the class of the
identity map,
ii)
(βoά)** = a**oβ**,
iii)
iv)
(«! 4- α 2 ) * * = &** + ^ * * >
α** is natural, i.e.,
(gAi-A)*0**** = a**o(g/\lB)*
for any map g: Y-+X.
2. 2. The mod q /z-cohomology theory [5], h{
is defined by
, A; Zq) = hi+2(XxMq,
hι (X Zq) = hi+2 (X A Mq)
Xx*[jAxMq),
for all i.
For a map / : Z - > F , A*(/; Zq) is defined by
«•(/; Zq) = (fAlM)*,
M=Mq,
and, for the sake of simplicity, denoted sometimes by
The suspension isomorphism
σq: h* (X
is defined as the composition
Zq)
Zq) and A(
Zq),
78
S. ARAKI and H. TODA
σq = (1XΛ T)*σ :
where T = T(S\ Mq). Since σ is natural, σq is also natural. With these
definitions h{ Z^) satisfies all axioms of a reduced cohomology theory.
Thus h{ Zq) becomes a cohomology theory.
Making use of maps (1. 6) we put
Pq
and
(or p) = (lΛπqrσ2:hi(X)^hi(X;
8Qf0 (or δ) = σ
Zq)
#
which are natural and called as the reduction "mod #" and the Bockstein
homornorphism respectively. The following relations are easily seen.
(2.2)
σqpq = ρqσ, 8qoσq=—
σSqo
and Sqoρq
= O.
From the exact sequence of h associated with the cofibration
and the definitions of pq and δ^0 we obtain the following exact sequence
(2.3).
^hi(X)-^hi(X;
Zq)^Xhi+1{X)-^~
,
where q denotes a homomorphism to multiply every element with q,
from which follows the exact universal coefficient sequence
(2.4)
0-^hi{X)®Zq-^hi{X\
Zq)-^Tor(hi+1(X),
Zq)->0
for all ί, which is natural, and the natural maps p' and δ' are induced
respectively by p and δ.
Put
δ ί i f = p f ϊ M : ί ' ( ^ ; Zq)-»ni+1(X; Zr)
for q, r>l, In case q = r, putting Sq = Sq qy we call it the "mod q" Bockstein homomorphism. The following relations follow from (2. 2) and the
definitions.
(2.20
σr 8
q r
= - Sq>r σq
andS
r s
S
q r
=O.
2. 3. The following propositions follow from (2.1) and Theo. 1.1.
Proposition 2.1. The groups h^X; Zq) are Zq-modules
(mod 4) and Zzq-modules if q=2 (mod 4).
if
MULTIPLICATIVE STRUCTURES IN MOD q COHOMOLOGY THEORIES I.
79
Proposition 2. 2. / / ??** = 0 in hy then h^X; Zq) are Zq-modules for
q>l.
any
As an example, we consider Atiyah-Hirzebruch if-cohomology theory
of complex vector bundles.
Theorem 2.3. Let a<={Sn+ry Sn}y then a** = 0 in K if rφO.
K*(X; Zg) are Zq-modules for any X and q>l.
Thus
Proof. Tensor products of vector bundles defines a multiplication
which induces natural isomorphisms : ^ # '(Z)(g)^ Λ (S Λ ) ^ Ki+n(XΛSn).
Via
the naturality of these isomorphisms it is sufficient to prove the triviality of a*:Rn(Sn)->Kn(Sn+r).
By Serre [8], {Sn+ry Sn} is finite for
n
n+r
rΦO. On the other hand, K (S ) = 0 or Z. Thus α* = 0. q.e.d.
The above theorem does not hold for /ΓO-cohomology theory of
real vector bundles. For, it is known [1], [9], that
KO-2(S°
Z2) = K0°(M2) ss Z 4 .
2.4. Take groups of coefficients Zg, Zr> and let a be an integer
such that
a q=0 (mod r).
Put af=aq/r
1
/
ά\S =a
and let a> a' :S1-+S1 be the maps defined as in 1.1.
1
1
1
and let a CS :CS ->CS
Put
be the canonical extension of a.
Then the map
(2.5)
with a\Sι=a'
a\Mr->Mq
and πgoά = Saoπr
is well-defined and continuous. The map a induces a homomorphism
5** = (lΛδ)*: hi+2(XΛMq)-+hi+2(XΛMr)
which is denoted by
(2.6)
* * : # ( * ; Zq)-»ft{X\ Zr).
We see easily
(2. 7), 0) 0* = 0.
If q = ry
1* is the identity.
ii) ^^p^ (Λ:) = a p, (Λ) /or x e
iii)
δrfoΛ*W = («ϊ/^) Sff.oW / ^ x^h'(X;
Zq\
iv) tf* 15 natural\ i.e., f*°a* = a*°f* for any map f.
We have
(2. 8 ) . 77*0 map f : Mr-^Mq
is homotopic
to zero.
Thus
r* = 0:hi{
Zq)
80
S. ARAKI and
H.
TODA
In fact, we define a homotopy rθ :Mr-+Mq between F and a constant
map as follows. Representing each point of Mr—S1 (or Mq—Sx) by
(x9t), χ(=S\ 0 < / ^ l , with the relations (#, 1) = * and (x,0) = r(x) (or
= q(x)), we put
rθ(χ) = (x, θ)
for
and
'ώfl-
Then rθ is a homotopy between ro = r and ^ = 0 (constant map).
Now let
be a mapping cone of the map ϊ:Mr-*Mrq.
For the map r:Mrq->Mqy
the composition f oΐ : My-^M^ is homotopic to zero by (2. 7), i) and (2. 8).
Thus the map r is extended over a map
using the above homotopy. We have
(2. 9). i? is a homotopy equivalence.
In fact, R is a deformation retraction by regarding Mq as the subspace S1UqCS1 of C(ϊ) as I | s i = ^.
Making use of the equivalence i?, we have the following exact
sequence (2.10) associated with the cofibration
X/\Mrq->X/\C(ΐ)
-*XΛSMr:
(2.10)
-. - # ( * ; Zg)-ϊ±»h'(X; Zrq)-^fr(X;
Zr)^
2.5. Compair α + 6 and aΛ-b in {Mr, M J . From the exactness of the
{S2, MJ - ^ {Mr, MJ — % {S1, M J and from the fact
2
we see that a + b — (ά + b) is in τr*{S , MJ which is
2
generated by iqηπr. If q or r is odd, then π?{S , Mq}=0. Therefore
we obtain
sequence
Proposition 2. 4. If q or r is odd or if y** = 0 in h, then a* + b*
= (a + b)* as natural maps\hι{
Zq)->hι(
Zr) in particular, a*(x)
l
=a x, x^h {X\ Zq\ when q = r.
(2.7), i), (2.8) and Prop. 2.4 show the following
Proposition 2, 5. βiven a homomorphisrn f:Zq->Zr
and let a be an
MULTIPLICATIVE STRUCTURES IN MOD q COHOMOLOGY THEORIES I.
81
integer such that f(t) = at (mod r). Then, under the assumption of Prop.
2.4, a* is independent of the choice of a, i. e., we may write
/* = **:#(
Z,)->ί'(
Zr).
Theorem 2. 6. // q is prime to r, then we have the isomorphism
(1*, 1*):/*' (X; Zgr)~H*(X;
Zg)®h\X;
Zr).
Moreover, if η** = 0 in hy the inverse of (1*, 1#) is
(ur)* + (vq)* = u.r* + v-q*:hi(X; Z,)®h'(X; Zr)^hl{X\
Zgr),
where u und v are integers such that ur+vq = l.
By assumption q or r is odd. Let q be odd. For l^ih^X; Zqr)
-+h'(X; Zq) and r#:h*(X; Zg)-^hi(X; Zqr)> l*°r* = r is an automorphism of h*(X; Zg) by (2.7), i), Props. 2.1 and 2. 4. From the exactness
of the sequence (2.10) the first assertion of the theorem follows. The
second assertion is straightforward from (2.7), i) and Prop. 2. 4.
When q is a multiple of r, we put
(2.11)
p, i r = l # : # (
Z,)->/?( ; Z r ) .
2. 6. The following theorem gives a condition for to split the exact
sequence (2. 4).
Theorem 2.7. If q^2 (mod 4) 6>r ί/ ?7** = 0 m h, then the sequence
(2. 4) splits:
Proof. Since Tor (hi+1{X\ Zq) is a Z^-module, it is isomorphic to
the direct sum of cyclic groups Fk (the set of indices {k} may be infinite
[6]). Let a be a generator of /^ and let 5 be the order of a. It is
sufficient to prove the existence of an element y of h{{X\ Zq) such that
/f
sy = 0 and 8 γ = a. Put t = q/s. By (2.7) we have the following commutative diagram :
K'(X) -^> H'(X; Zs) i
i'
Tor (h
),Z,)-»0,
•\vhere i is the inclusion, J,et β<Ξh'(X; Z?) be chosen such as β'β = a,
82
S. ARAKI and H. TODA
Put y = t*β. Then δ'γ = α. If ^ = 0 in A or if 5 ί 2 (mod 4), then
sβ = 0, thus sy = 0. The remaining case is that s=2 (mod 4) and # = 0
(mod 4), then t is even. By Theo. 1.1, we have
sΎ = t*(sβ) =
= t*psσ-*(lΛivy*β = t-pqσ-\l/\i ηYβ = 0 ,
since t is even and /iy = 0. q. e. d.
Corollary 2.8. (2.4) splits for h = K:
i
K<(X; Zg)^K (X)®Zq®
i+1
Tor (K (X), Zq)
for any X and q>l.
By a parallel proof to that of Theo. 2. 7 we obtain
Theorem 2. 9. The sequences (1. 7) and (1. 70 s/>/# // #ΞJΞ2 (mod 4):
{XAMqy W} « {S2X, T^}®Z, ΘTor({SX,
{F, XΛMJ ^ {Y, SX}®Z, θ Tor ({F, S2X}> Zq).
3. An axiomatic approach to multiplications in mod q cohomology
theories
3.1. A cohomology theory h is said to be multiplicative, if it is
equipped with a map
(3.1)
μ
: h£(X, A)®M(Yy B)->hi+J(Xx Yy XxBΌ Ax Y)
for all iyjy which is
(Mi) linear,
(M2) natural (with respect to both variables),
(Λf3) has a bilateral unit l<=h°(S°, *), i.e., μ(l®x) = μ(x®l) = x for
any x^h{(X, A),
(M4) compatible with the connecting morphisms. cf., Dold [5], p. 6.
If μ is associative, i. e., satisfies
(M5)
(where we used 1 to stand for an identity map of a group and such a
kind of usage of 1 would not give rise to any confusion with the unit
°, *)), or if μ is commutative, i.e., satisfies
T*μ{x®y) = ( - ϊ)ijμ{y®x)
MULTIPLICATIVE STRUCTURES IN MOD q COHOMOLOGY THEORIES I.
l
83
j
for x^h {XyA) and y<=h (Y, B\ where T:YxX-^XxY
is a map
swiching factors, then we say that μ is an associative, or commutative,
multiplication.
To give a multiplication μ in h is equivalent to give a multiplication in h
+
(3.2)
μ : #(X)<g)#(r) -^# >(XΛ Γ)
for all ί, y, such that they transform naturally to each other by Ithe
passages from h to h and converse. About the multiplication (3.2) the
axiom (M4) is replaced by
(MJ) the compatibility with the suspension isomorphism σ:
σμ{x®y) = (l/\T)*μ(σX®y) = (-1)* μ(x®σy)
for άegx = i, where
T=T(YyS1).
By the reason of this equivalence our discussions are limited only
for multiplications (3.2) in h.
3. 2. Let h be a multiplicative cohomology theory with a multiplication μ. The multiplications
μR:h'(
(3 3)
Z,)®#( ) # (
Z
^ O β W; z ) ^ (
9
)
z)
f o r a U
are canonically induced by requiring that the following diagrams should
be commutative:
K'(X; Zq)®tt{Y)
Ϊίi+J(XΛ Y; Zq)
—
Λ Y)
q
Zq)
- ^ ^ ( I Λ 7 ;
Zq)
YΛMq).
As is easily checked, μR and μL satisfy the following properties :
(Hj) linear
(H2)
(H3)
(H4)
natural
1 is a right unit for μR and a left unit for μL
compatible with suspension isomorphisms in the sense that
84
S. ARAKI and
H.
TODA
σqμR(x®y)
= (1Λ T)*μR{σgx®y)
= (-1)* μR(x®σy) ,
σqμL(x®y)
= (1Λ T)*μL{σx®y) = ( - 1
(i/5) compatible with the reduction mod # m £/z# sense that
(H6)
compatible with the Bockstein homomorphisms in the sense that
SμR(x®y) = μ(8x®y), δμL(x®y) = (-i
= μR(8gX®y),
SgμL(x®y) =
(-iy
for
If μ is associative, then the following associativity
(H7)
holds
and it μ is commutative then the commutativity
(H8)
holds for x^hHX) and y€=fc(Y; Zq\ where
T=T(Y,X).
If 77*^ = 0 in h, then the following compatibility with the homomorphisms of coefficients holds :
(H9)
f*μR = μ*(/*®l)> f*μL = μz.(l®/*) >
where / # : A**( Z^)->^f'( Z r ) is induced by a hornomorphism / : Zg->Zr.
For general case (i/9) holds after replacing /# by #* (<z, an integer).
3. 3. Let h be a multiplicative cohomology theory with an associative
multiplication μ. We shall discuss multiplications
(3.4)
μe .h'iX; Zq)®y(Y\
Zq)->ni+>\XΛY;
Zq)
in fo( Zq) by postulating the following properties:
(Λo) μq is a multiplication, i. e., satisfies (MJ, (M2), (M3) and (Mi)
for the cohomology theory h( Zq)
(Λj) compatible with μR and μL through the reduction mod qy i. e.,
μR = μq(l®pq)
and
μL =
μq(pq®l);
MULTIPLICATIVE STRUCTURΉS IN MOD q COHOMOLOGY THEORIES I.
(Λ2)
8$
δ* is a derivation (in the graded sense), i. e.,
for deg x = i
(Λ3) "quasi-associative" in the sense that, if at least one element of
{x> y> z} is in pq-irnages, then the associativity
μq(μq(x®y)®z)
=
μq(x®μq{y®z))
holds.
(Λo) is a minimal requirement to call μq as a multiplication. (ΛJ
means a compatibility of μq with μ. In fact, (Λx) and (Hs) imply that
(ΛQ μq is compatible with μ through the reduction mod^ in the
sense that
Denote by lq the bilateral unit of μq, then we have
Proposition 3.1.
/ / a multiplication μq satisfies (Λ^, then
(A10
μq.
l
From (Λx) and (H3) we see easily that pq(l) is a bilateral unit of
Then from the uniqueness of bilateral units follows (Λi7)-
Proposition 3. 2. // a multiplication μq satisfies (Λ")> then h*(X
is a Zq-module for any X.
For any xϊΞh*(X;
Zq)
Zq\
q x = μq(lq®q-x) = μq(q-ρq(ϊ)®x)
= μq(pq(q ]-)®x) = 0
because pq(qΊ)
= 0 by (2.3). Thus Prop. 3.2 was obtained.
From Prop. 3.2 we see that, if μq satisfies (ΛiO> the exact sequence
of fo( Zq) associated with the cofibration
breaks into short exact sequences
(3.5) 0->
for any X and L
86
S. ARAKI and H. TODA
Postulation (Λ2) and (Λ3) are necessary to get theorems of uniquenesstype. A multiplication μq satisfying (Λo), (ΛJ, (Λ2) and (Λ3) is called
an admissible multiplication.
3. 4. Assuming the existence of μq satisfying (Λ")> put
κ2 = π*σq2lq£Ξh2(Mq
(3. 6)
Zq).
By the exactness of (3.5) for X=S°, π* is monomorphic. Hence /c2Φθ.
We shall prove that there exist an element /^eA^M*; Zq) such that
(3. 7)
8q/c1 = κ2 and i*^ = — σqlq .
Let
πf:
MqΛMq->(MqΛMq)/(S1ΛS1)
be a map collapsing S ^ S 1 to a point. The map
lAi:MqAS1-^MqAMq
(or
iAl:S1AMq^MqAMq)
induces an injection
ί 1 : S 8 = S 2 ΛS 1 ->(M,ΛM,)/(S 1 ΛS 1 )
(or i2:S3 =
S1AS2->(MqAMq)/(S1AS1))
such that the commutativities
7r'(lΛ0 = *Ί(tfΛl)
hold. Putting ik(S3) = Szky k = l
structure
and TΓ^IΛI) = f2(lΛτr)
or 2, we obtain
the following
cell
(Λ^ΛMΛ/C^ΛS 1 ) = SJ VSi U , 1 + , 2 e*,
where ^ : S 3 - ^ S ? , ^ = 1 or 2, is a map of degree q. Let /£, k = l or 2,
be the map ik considered as the maps into S?VS|, and
the map collapsing S?, / φ ^ , to a point such that
PA = I s 3 .
Then
since pkQk^Q- Thus, by the exact sequence of ^ associated with the
coίibration
MULTIPLICATIVE STRUCTURES IN MOD q COHOMOLOGY THEORIES I.
87
there exists an element
3
1
1
such that j*κ =
κ^h (MqAMq/(S AS ))
3
3
pfσ l-p$σ l.
Put
Kl
= Λ e ί ? ( M f f Λ M f ) = h\Mq Zq),
which satisfies (3.7) as will be proved in the following way :
*ri[*j*ic = (TΓATΓ)*σ 4 l - * 2 ,
The choice of ^i is not unique. Nevertheless we fix κ1 once for all.
We remark that in the choices of κ1 and κ2 we did not use any special μg.
3. 5. The next proposition gives a necessary condition for the existence of an admissible multiplication in case of q=2 (mod 4) which is
sufficient for the existence of a multiplication μq satisfying (Λx), c. f.,
Cor. 5.6).
Proposition 3. 3. If q=2 (mod 4) and there exists a multiplication
μq satisfing (Λ")> then
p,i7*(l) = 0 and (vπq)** - 0 in h.
Proof.
^ Λ1 = (^ lΛf)*Λi = (/i7τr)*ιc1 by Theo. 1.1,
= (IΪTΓ)*!**! =
=
-(ηπ)*σglg
-<rqπ*(y*lq).
On the other hand ^-^ = 0 by Prop. 3.2. Thus
σ^*(i2*lf) = 0.
Now σ ? and π jj5 are monomorphic (by (3. 5)). Hence
By (Λί'), 7*l ί = i7*σί(l) =P ί 7 * ( l ) . Thus
Next,
For any
88
S. ARAKI and H. TODA
{ηπq)**χ = (yπq)**μ(x®l) = (ηπ,)** n{a
= μ{σ-2X®(Sπq)*r}*σ2l)
= μ(σ-2X®T*(lΛπq)*
where T=T(Mqy
T*(σ2y*l),
S1) and 7 > 7XS1, S2). Since T? = l,
(vπg)**X =
μ{σ-2X®T*(l/\πq)*σ2η*l)
= 0. q.e.d.
3. 6. Next proposition can be viewed as a special kind of Kϋnneth's
theorem.
Proposition 3. 4. // a multiplication μq satisfies (Λ")> then for any
X and x^h*(XAMg;
Zq) it can be expressed uniquely as a sum
X = μq{x1®K1) + μq(x2®K2)
with x^fr-^X;
Proof.
Zq) and x^h^X;
Zq).
Define a homomorphism
kifriXΛS1;
Zq)->fr(XΛMq;
Zq)
by putting
k(y) =
(-l)iμq(σ-ιy®fc1).
By an easy calculation we see that
^ ) * ^ = an identity map,
i. e., k gives a splitting of the exact sequence (3.5). Thus, for any
i
1
x^h (XAMq;
Zq) two elements y^h'iXΛS ;
Zq) and / G A ^ Z Λ S 2 ; Zq)
are determined uniquely so as to satisfy
Put
x1 = (-lYσ^y
and
2
x2 = σ~ y'.
Then, by (3. 6) and (3. 7), we get
X = μq(x1®K1)-\-μq{x2®κ2)
.
The uniqueness of x1 and x2 follows also from the exact sequence (3. 5)
and the definitions of κ1 and κ2.
3. 7. Let μ be an associative and commutative multiplication in h.
We fix μ once for all throughout this paragraph and shall discuss
MULTIPLICATIVE STRUCTURES IN MOD q COHOMOLOGY THEORIES I.
relations between different admissible multiplications μq, μqy μq
To simplify notations we put
89
etc.
etc.
y
For two μq and μ'q we see by (Λx) that
(3.8)
xΛy = xΛ'y
if either x or y is in pq-images.
In particular,
(3.9)
δgxΛy = δqxΛ'y
and
xΛδqy = xΛ'δqy
for any x and y.
Thus, by (Λ2) we obtain
(3.90
8q(xΛy) = 8q(xΛ'y)
for any x and y.
Also, by (H8) and (ΛJ we obtain
(3.10)
T*(xAy) = ( — l)ijyAx
where άegx = i and degy=j.
if either x or y is in pq-images,
By (3. 6)-(3.10) we obtain
(3.11)
KiΛtcj = KiΛ'tcj
if i = 2 or j = 2,
(3.12)
T^KiΛfcj) = Kj/Mti
if i = 2 or j = 2.
3.8. By Prop. 3.4 every element ^ G A ^ I Λ M ^ Λ M ^ ; Zq) can be
expressed uniquely as
x=
with
x^h'-^X;
£ 3
Zq\ x2> x3£Ξh - (X; Zq\ x^ft-\X\
Zq).
Put
(3.130
T*(*i Λ icd = (<*i Λ *i) Λ tc, + (a2 Λ« 2 )Λ«!
+ (Λ 3 Λ /ci) Λ κ2 + (α4 Λ κ2) A κ2
with ^ G A ° ( S ° ; Zq\ a2f a^h-'iS0; Zq\ α 4 e ^ 2 ( S 0 ; Zq\ Apply (iΛl)*
on both sides of (3.130, then, by (3. 6), (3. 7) and (3.10), we obtain
σqlqA/c1 = ("-σga1)Aκ1 + σgaΛAκ2
Thus, by Prop. 3.4, we see that
aλ = —lq
and a3 = 0 .
90
S. ARAKI and H. TODA
Similarly, applying (lΛi)*
o n
b o t h
sides of (3.130, we see that
Finally, making use of (Λ3), we get
(3.13)
JΓ*(*I Λ * i) =
- κx A κt + a(μq) A (κ2 A κ2)
with a(μq)^h-2(S°;
Zq). Apply 8q on both sides of (3.13), then, by (Λ2),
(3. 6), (3.7), (3.12) and Prop. 3. 4, we see that
(3.14)
δqa(μq) = 0.
a(μq) is characteristic of
Next, put
(3.150
with
μq.
* i Λ '*! = (b, A *O Aκ, + (b2 A κ2) A κx
b^h^S0;
Zq\ δ 4 eA' 2 (S°; Zg).
Zq\ b2> b3^K-\S0;
Apply (iΛl)* on both sides of (3.150, then, by (3.6), (3.7), (3.8) and
Prop. 3.4, we see that
bλ = lq
Similarly, applying (lΛi)*
o n
and
b3 = 0 .
both sides of (3.150, we see that
Thus, making use of (Λ3), we get
(3.15)
κ1Aκ1 — κ1A/κ1 = b{μqy
μ'q)Λ(κ2Aκ2
with b(μq, μ'q)^ϊi-2(S0; Zq). Apply δ^ on both sides of (3.15), then, by
(Λ2), (3.7), (3. 90 and Prop. 3. 4, we see that
(3.16)
8qb(μqy
μ'q) = 0.
By (3. 8) and (3.15) we obtain
(3. 17)
b(μq, μ'q') = b(μq, μq) + b{μ'q , μ'q') .
Apply T * on both sides of (3.15) and make use of (3.13)
by Prop. 3. 4 we get the relation
(3.18)
a(μq) - a{μ'q) = 2b(μq,
3. 9. Here we state the following
μq).
then,
MULTIPLICATIVE STRUCTURES IN MOD q COHOMOLOGY THEORIES I.
Lemma 3. 5. Let μq be an admissible multiplication in Iϊ(
φ
and α e ^ ( S ° ; Zq\ then
Proof.
91
Zq).
If
By (3.10) we have
where T=T(S\X).
that T(S°, X) = lx.
Via the identification XAS° = S°AX=X,
we see
Thus T* is an identity map, and the lemma follows.
The next theorem shows that b(μq, μq) measures the difference of
μq from μq.
Theorem 3. 6. Let μq and μq be admissible multiplications in h(
Then
,
Zq).
μq)A(8qxA8qy)
for any x&fc(X; Zq) and ytΞhj(Y\ Zq).
Proof. In case y = κλ:bγ (3. 9)-(3. 90 we obtain
from which, making use of (Λ2), (Λ3), (3. 8) and (3.10), we get
( - l) f (x A '(* 2 Λ ^ J - x A {κ2 A κx))
= SqxA(fc1Afc1
= (SqXAb(μq,
,
— κ1A'fc1)
μq))A{κ2AfC2)
μ'q)ΛSqx)Λ(/c2Λκ2)
by Lemma 3.5. Here apply (1XAT)*> T=T(Mq, Mq)y on both sides of
this equality. Making use of (Λ3), (3.11) and (3.12), we obtain
((b(μqy μ'q)ΛδqX)Λκ2)Λκ2
= (—
1)*(xA/κ1-xΛ«i)Aκ2.
Then, from the uniqueness of the expression of Prop. 3. 4 follows
This shows the theorem in case y^/c^
The theorem for x = κλ can be proved similarly as above by deforming the formula
92
S. ARAKI and H. TODA
and we obtain
(*2)
KiΛy-^Λ'y
= b(μq>
μ'q)Λ(*2Λδgy).
Now we shall discuss the general case. By (3. 9)-(3. 90 we have
Decompose the both sides into four terms by (Λ2), and apply
T=T(Y, Mq), on the both sides. Remarking that we can drop many
brackets by the quasi-associativity (Λ3) we obtain
Thus
= 0.
Rewrite the first three terms by making use of (*1) or (*2). Then, by
using Lemma 3. 5, we see that the first and second terms cancel to each
other, and obtain
Making use of Prop. 3. 4 twice, we obtain
/
-xA y)
= O. q.e.d.
3.10. The following theorem shows that a(μq) measures the deficiency of μq from the commutativity.
Theorem 3.7. Let μq be an admissible multiplication
Then
for any x^h'iX;
Zq) and y(E&(Y; Zq\ where T=T(X9
in h{
Zq).
Y).
Proof. Put μq\x®y) = { — l)ij T*(yAx)y then it is a routine matter
to see that μ'q is also an admissible multiplication. (3.13) shows that
MULTIPLICATIVE STRUCTURES IN MOD q COHOMOLOGY THEORIES I.
93
κx A fκx = /c1Aκ1 — a(μg) A (κ2 A κ2) .
Hence the theorem follows from Theo. 3. 6.
Theorem 3. 8. Let μq be an admissible multiplication in h( Zq) and
any b<=fι-2(S°; Zg)nS~ι(0) given. If we put
l
j
for x^.h {X\ Zq) and y<=h (Y; Zq\ then μq is also an admissible multiplication and b(μqy μq) = b.
(Λ2).
Proof. It is straightforward to see that μq satisfies (Λo), (ΛJ and
By a simple calculation we see that
(*3)
{xA'y)A'z-xA'(yA'z)
= ((xAy)Az-xA(yAz))
where j=degy.
If x is in p^-images, then
bAx=xAb
by Lemma 3. 5. If y or z is in p^-images, then
Thus, if x or y, or z, is in p^-images, then the second term of the left
side of (*3) vanishes, and the first term also vanishes by (Λ3) for μq>
i. e., (Λ3) for μ'q was proved, q. e. d.
In the formula (*3), if b is in p^-images, then
bAx=xAb
by a similar proof as in Lemma 3. 5. Thus we obtain from (*3) that
(3.19) if μq is an associative admissible multiplication and b is in pqimageSy then the multiplication μq defined as in Theo. 3.8 is also associative.
3.11. From Theos. 3. 6, 3. 7, 3. 8, (3.18) and (3.19) we obtain the
following corollaries.
Corollary 3. 9. Let μq and μq be two admissible multiplications. The
following conditions are equivalent.
94
S. ARAKI and
H. TODA
ϋ)
b(μq, μ'q) = 0,
iii)
μq coinsides with μ'q for the case of
X=Y=Mq.
Corollary 3.10. / / there exists an admissible multiplication in
h( Zq), then admissible multiplications in 1i{ Zq) are in a one-to-one
correspondence with the elements of ^~2(S°; Zq) Πδ^XO).
Corollary 3.11. If q is odd and admissible multiplications exist in
h( Zq), then the correspondence μq-^a(μq)
is bijective, and there is just
one commutative multiplication {which corresponds to a{μq)=0).
Corollary 3.12. / / q is even and admissible multiplications exist in
h( Zq), then either there is no commutative one, or commutative ones are in
one-to-one correspondence with the elements of Tor (h~2(S° Zq) Π δ^Xo)* Z2).
Corollary 3.13. / / there exists an associative admissible multiplication in h{ Zq) and h~2(S°; Zq) = pq(h-2(S0)),
then every admissible
multiplication in ϋ{ Zq) is associative.
3.12. Assume that q and r are relatively prime integers, u, v are
integers such that ur-\-vq = l, and 17** = 0 in h or qr is odd. Given a
multiplication μqr in ΐι{ Zqr), we define multiplication μq in fι(
Zq)
and μr in h{ Zr) respectively by the formulas
μr{x®y) = Pqr> rμ
where («r)* h*( Zq)-+ti*(
Zqr) and (υq)*: X*( Zr)->ϊϊ*(
Zqr). If
μqr satisfies (Λί7), then it is straightforward to see that μq and μr are
multiplications satisfying (Λ") by (2. 7) and Prop. 2. 4. Given multiplications μq and μr, we define a multiplication μqr in $( Zqr) by
(3. 21) At^(ΛrΘj') = (ur)*μq(pqrt
x®pqr, qy) + (vq)*μr(pqrt rx®pqrf
q
y).
τ
Also in this case, if μr and μq satisfies (Λ")> then μqr becomes a multiplication satisfying (Λί7).
Theorem 3.14. Under the assumptions that q and r are relatively
prime, and 77** = 0 or qr is odd, the correspondences μqr-+(μq> μr) and
(μq, μt)->μqr, defined by (3.20) and (3.21) respectively, are bijections of
multiplications satisfying (Λ(7) which are the inverses of each other. μqr
satisfies (ΛJ, (Λ2) or (Λ3) if and only if μq and μr do so.
Proof. The first assertion follows from a simple calculation to check
that the two correspondences are the inverses of each other (making
MULTIPLICATIOE STRUCTURES IN MOD q COHOMOLOGY THEORIES I.
95
use of Props. 2. 4, 2. 5 and (2. 7)). The assertions for (Λ^ and (Λ3) are
also easily checked.
To prove the assertion for (Λ2), first we remark that, by (2.7), ii)
2
and iii), when as=0 (modt) and as /t = 0 (mod t) the following diagram
is commutative:
;zs)
k
8.
In the following calculations the above commutativity is used.
Assume that μqr satisfies (Λ2), then
8rμr = δrpgr,r
± (υq)* ®Sqr (vq)*)
= q
That is, μr satisfies (Λ2). Similarly μq satisfies (Λ2).
Next, assume that μq and μr satisfies (Λ2), then
8qrμqr - 8qr(u
{q 1*8^ ® 1*)
{t^q)*μr{l*®qΛ*8g
r
= μ>gr(δgr®l)±μgr(l®δqr) ,
i. e., μqr satisfies (Λ2). q. e. d.
4. Stable homotopy of some elementary complexes. I.
4.1. The results in the following table are well known.
w+
n
{S S S }
generators
ί<0
ί=0
ί= l
ί=2
ί =3
i=4, 5
0
Z
z2
z2
Z24
0
1
V
V
96
S. ARAKI and H. TODA
From (1. 7) and (1. 7') we have the exact sequences
\ Sn}(
, Sn}
(4.10
Λ+
Sn+1}®Zq
%
*', S*M,}
(4.1) When q is even, there exist elements
τ}<={Sn+3y SnMg} such that i*η = ηi=η and
η^{SnMqy
Sn}
and
(4.1) is clear from the exactness of (4.10 for i = 3. We see also
that the groups {SnMqy Sn} and {Sn+3y SnMq} are of order 4 when q
is even.
(4.20
If Q=2. (mod 4), then 2η = v2π and 2y = iv2.
Because: making use of Theo. 1.1, we see that
2η = 2η + (q—2)η = η(qΛ) = ηiηπ = v2π ,
and
From (4.20 and Theo. 2. 9 applied to (4.10 we obtain
(4.2). The groups {Sn+i~3Mqy Sn} and {Sn+£, SnMq}
to the corresponding groups in the following table:
i=2
i=3
0
0
Zq
0
q=0 (mod 4)
0
zq
zq
z2
z2
0
n+i
z
generators of {S ~ Mqt
M
S>
generators of {SM+», SnMς}
4. 2.
(4. 3)
i=l
q : odd
q=2 (mod 4)
are both isomorphic
Zcq,2θ
Z 2 + Zcί,24)
z,
Z 2 + Zc?,24)
2
π
ηπ
V,7] π
ηfj, vπ
i
iη
7), iv2
ηv, iv
For even q> we use the following notations
Vl
= iη,
v2 = yπ£Ξ {S»+1 Mq, S"Mq} ,
n+2
n
(4.4) When q=0 (mod4), there exists an element y3tΞ{S Mq,
S Mq}
such that πvsi = V arid 2^3 = 0. Thus we may choose η and η such as
η = πv3 and v = v^i> theft
MULTIPLICATIVE STRUCTURES IN MOD q COHOMOLOGY THEORIES I.
Vi = (iπ) y3
97
y2 = y3 (iπ).
and
By Theo. 2.9, there exists an element y3 such that y3i = i*y3 = η and
2^3 = 0. Then πη3i = πη = rj. Changing η by 7r*?3 if necessary, we see that
(4. 4) holds.
From (1. 7) and (1. 7') we have the following exact sequences:
*'*
π*
n+i+
Tor ({S
n
\
S Mq}>
, S
W+2
},
We obtain
Theorem 4.1. The group {Sn+iMqy SnMq)
responding group in the following table:
q : odd
0
generators
q = 0 (mod 4)
0
generators
q=2
(mod 4)
0
generators
is isomorphic to the cor-
ι=-l
ί==0
i=l
i=2
zq
zq
0
Zcq. 24)
iπ
1
ivπ
zQ
Z2 ~\~ Z2 ~\~ 2*2 Z2 + Z2 4" Z2 + Z(qt 24)
iπ
1, iηπ
zq
Z iq
Z2+Z2
Z 2 + Z 2 + ZCΪ,24)
iπ
1
Vl> V2
??i 2 , ?722> IVJΓ
7]i,V2>
τ?i2, 7]22> Vi* ivπ
irfπ
Proof. In case #ΞJΞ2 (mod 4), the above two sequences split by
Theo. 2.9, and the results follow easily from (4.2). In case q=2
(mod 4): for / = 1,2, combining the above two sequences we see that
the sequences split, then we have the results; for i^~ 1, the proof is
obvious; for ί = 0, the results follows from Theo. 1.1. q. e. d.
Corollary 4. 2. (i) In case q=Q (mod 4 ) : {Sn+iMq, SnMq] are multiplicatively generated for i^2 by iπ, y3 and ivπy i.e., putting δ = /τr we
have the relations:
iη2π = 8η3 δy3 δ,
η2 = δy3 8y3,
η2 = y3
3
δ,
δδ = S(ivπ) = (ivπ) δ = 0 .
(ii)
case
q=2
(mod 4 ) :
, SnMq)
are
multiplίcatίvely
98
S. ARAKI and H. TODA
generated for i^2 by iπ ( = δ), y19 v2 and ivπ with the relations:
δδ = Sv, = y2δ=δ(ivπ) = (ivπ)δ = 0 ,
Ύ]χ δ = δη2 = tηπ = q 1,
ηxV2 = η2ηx = 0 ,
Proof. The proof is easy except the relations on ηxη2.
By (4.3) VlV2 = iηyπ. By (4.2)-(4.20 2y = (q/2)v2π. Then,
This implies that
?7?7 = 3^z/ m o d
Thus
V!V2 = 0 mod 12(ιW),
where 12(fW) = 0 if ^ = 2 (mod 4). q. e. d.
4.3. We shall see that MgΛMq is homotopy equivalent (in stable
range) to the following mapping cone
(4.5)
Nq
g
where
Si)v(Sπ):SMq^S'^S2->S3dSMq
0 (constant map)
if
^^
if
# ΐ 2 (mod 4).
We denote also by Nq a subcomplex of Nq obtained by removing
the 3-cell SMq-S\ i.e.,
(4.50
S2U~gC(SMq),
Nq =
where
3
2
_ ί v(Sπ): SMq ->S ->S
g
~
Obviously
10
N^ = iV^USM^ and
if
q=2 (mod 4),
if
tfΐ2
(mod 4).
NqΓ\SMq = S2.
The cell structures of Nq and 7V^ can be interpreted as follows:
(4.6)
(i) if qm2 (mod 4),
Nq = SMqVS2Mq
and
Nq =
and
Nq = (
(ii) // q~2 (mod 4),
[Je4
S2VS2Mq;
MULTIPLICATIVE STRUCTURES IN MOD q COHOMOLOGY THEORIES I.
where eA is attached to S2VS3 by a map representing the sum
VΪΞ{S\
S2} and S2q = qΛ3tΞ{S3, S3}.
99
of
We use the following notations:
(4.7)
j:NqaNq>
the
io:S2c:Nq,
tQ:SMqaNqy
3
3
t1:S czNq,
iλ :S aNqy
3
p : Nq-^S y
2
inclusion)
the inclusions)
the inclusions
the map collapsing Nq :
τro:Nq-^S Mqy
πo:Nq-+S2Mqy
the map collapsing SMq or S2.
Hereafter, these mappings will be fixed as to satisfy the following
relations:
(4. 70 jiQ = ιo(Si), ~ix =ji1,
pt0 = Sπ> π0 = πj
and
τtjι = πQiλ = SH .
Lemma 4.3. Teere exists an element όί of {Nq, Mg/\Mq]
satisfying
the following three conditions:
(4. 8), (i). a is a homotopy equivalence, i. e., there is a {uniquely determined) inverse β^{MqΛMq,
Nq} of cc such that άβ=l and βcc=l.
(ii).
(iii).
aϊo = lMΛi,
thus &(lMΛi) = t0( 1 M Λπ)ά = τr0, thus
πβ=lMΛπ.
Proof. In general, a homotopy between / and g:X-^Y
gives a
homotopy equivalence h: Y{JfCX->
Y\JgCX such that h Y=lγ
and,
by callapsing Y, h induces a mapping h:SX->SX
homotopic to lSχ.
By Theo. 1.1, g is homotopic to qΛSM in stable range. On the other
hand MqΛNq is a mapping cone of q-lSM. Thus the lemma follows,
q. e. d.
Put
(4. 9)
a0 = δ i . e {S3, MqΛMq}
and
β0 = pβtΞ {MqΛMq>
S3} .
It follows from (ii), (iii) of (4. 8)
(4.90.
(lMΛπ)a0
= S2i
and
βo(lMΛi)
= Sπ.
Remark that
(4. 9")
if a'Ό and β'o satisfies (4. 90 then
ao—a'o = 0 or =(iΛi)v
and
where (iΛi)v = v(πΛπ) = O if q is odd.
/?0-/3o = 0 or
=v(πΛπ),
100
S. ARAKI and
For, α o - α S e ( l M Λ θ * (S3, SMq}
(ίcf. (4.2)).
H.
TODA
and
βo-β'Q<Ξ(lMAπ)*{S2Mqy
S3}
Lemma 4.4. (i). Let cc^{Ngy MqΛMq}
be an element satisfying
(4.8). Any element ά'<={Nq, Mq/\Mq} satisfies (4.8) if and only if
a' = a+(lMAi)Ύτr0+x(iΛi)vP
for some y<={S2Mq, SMq} ,
where x = 0 if q^2 (mod 4) and x = 0 or 1 // q=2 (mod 4).
(ii). For any α o e {S8, Mq/\Mq) and βo<= {MqΛMq, S3}
satisfying
(4.90 f*ere e#*sfo « G { ] V , M 9 AM ? } which satisfies (4.8) dwd (4.9).
Such an element a is unique if q is odd, a or δ+(tΛi)v 2 (S 2 π)τr 0 if q=0
(mod 4), a or ct+(iΛi)vP if q=2 (mod 4).
?
Proof, (i). Assume that a and ec' satisfy (4.8). There exists
γ'(Ξ {Nq, SMq} such that (l M Λ/)γ / = ά:/ — ά since ( I M A T Γ X ^ —ά) = 0. Then
s
(1MΛ/)Ύ I = (^-5)10 = 0. The kernel of ( l M Λ i ) * : ί ^ >
SM ff }->{SM g ,
M^ΛMJ is q{SMqy SMq) which vanishes if q^2 (mod 4) and is generated
by (Si)v(Sπ) if q=2 (mod 4). We have (Si)vpϊo = (Si)v(Sπ) by (4.7 7 ).
Thus (7/-Λτ(Sf)^)f0 = 0 for some X(<EΞZ if ^ Ξ 2 , = 0 if q^2).
Then
there exists γ E { S 2 M ? , SMq} such that yτtQ = rγ/ — x(Si)vpy and
/
0
2
Conversely, if a satisfies (ii), (iii) of (4.8) then so does at', cc and
oif induce the same homomorphisms of ordinary cohomology groups.
Thus cif is a homotopy equivalence if so is ά.
(ii). If q is odd a0 and β0 are unique by (4. 9"). Also a is unique
since {S2MqySMq} =0 if q is odd. Thus (ii) is obvious for odd q.
Let q be even and choose an element cίf satisfying (4.8). By (i)
and (4.1), any a" satisfying (4. 8) can be written in the form
with xy y> z<E.Z2, where S = (iAi)vp if q=2 (mod 4), and S = (lMAi)(Si)
y2(S2π)τr0 = (iΛi)y2(S2π)τr0
if q = 0 (mod 4).
By a calculation making use of (4.1), (4. 3) and (4.70 we see that
a'o
f
=a'Ό-\-x(iΛi)y,
where a^ά'I, and a j = a'fiλ. Putting β'0=pi5' and β'<!=pB" iβ' and β"
are the inverses of άf and ccf/ respectively), by a similar calculation we
see that
On the other hand
MULTIPLICATIVE STRUCTURES IN MOD q COHOMOLOGY THEORIES I.
101
Since oc" is a homotopy equivalence we obtain
Now, for the given a0 and β0 satisfying (4. 90, by (4. 9") we can put
a0 — aί = x'(i A i) ^
and
βo—βi = y'v(π A π)
with # ' , / e Z 2 . Since, for arbitrarily chosen jt, y and £, the element a"
satisfying (4. 8) and (*) exists uniquely by (i), if we put x = xr and y=y',
then the determined a" is the required element. Thereby z can not be
determined by given a0 and β0 but may have two possible values, which
corresponds to the conclusions of the lemma (ii). q. e. d.
4.5. Consider the groups {MqAMqf S2Mq} and {SMq, MgΛMg}.
From (1.17) and (1.170 we have the following exact sequences:
r9>
s*M9}®zg
„ SMg)®Zg
A. {MgΛMg, S*Mg}
{SMt, S2Mg} ,
'-$ {SMg, Mg/\Mg)
{SMg, S*Mg) .
By (4.2) and Theo. 4.1, (STΓ)* : {S3, S2Mq}->{SMq, S2Mq} is an isomorphism. The formula βo(lMΛi)=Sπ of (4.90 implies that β^Sπ)*'1 is a
right inverse of (lΛi)* Thus the first sequence splits. Similarly the
second sequence splits since αoaeCS2*")*1 ^s a right inverse of
Then it follows from Theorem 4.1
(4.10).
03ΞO (mod
{M ? A Mq, S2Mq}^{SMg,
Mq Λ M J ^
4)
Zq ~\~ Zq
generators of {M9 Λ Mq, 52M^}
generators of {SMg, M9 Λ M ? }
^ = 0 (mod 4)
ljfΛw, (S 2 ί)Λ» (S2ί)??(jr A π)
ljf Λ i, α o (57r)
ljfΛί, αo(S7τ), (ί Λ i)3?(Sπ)
Lemma 4.5. For each q>\, there exist elements α o =α o ? e
{S3, MgΛMg} and βo = βOtβe.{MgAMβ, S3} which satisfy (4.9') and the
following relations:
,. ...
(4.11)
Proof.
(i) (lMΛπ)T
= lMΛπ + (S2i)β0,
i-a(Sπ)
Let G be a subgroup of {M?ΛM,, S 2 M ? } generated by
102
S. ARAKI and H. TODA
(S2i)v(πΛπ) iί q = 0 (mod 4), or consisting only of zero iί q^O (mod 4).
Then, using an element β0 satisfying (4. 90, we can put
(#1)
(lΛπ)T*=a(lΛπ) + b(S*i)β0 mod G,
for some a, b<^Zq by (4.10).
Let us use the ordinary mod q reduced cohomology ίϊ*(
the generator gi^ίϊi(Mq;
Zq)> * = 1,2, such that
ί*Si = - σlq
Zq) and
2
and τr*(σ- lq) = g2,
i.e., gΛ=κ{ in the sense of 3.4. Then, by (3.7) and Prop. 3.3 8qg1=g2
and the four elements giΛgj^μg(gi®gj)
form a base of H*{Mq/\Mq Zq),
where μq is the reduced cross product.
From (# 1) we obtain the identity
Applying this to σ2g1=g1Λσ2lq,
of cohomology maps.
&Λft =
3
Since the class βf(σ lq)
for some x^Zq.
we have
a(giΛg2)-b-β$(σ*lg).
is integral,
It follows from (4.90 that
=
-X-<rg2
Thus, ΛΓ= —1 and
That is, α = 6 = 1, and
(#2)
(lAπ)T = (lAπ)-h(S2i)β0
mod G.
This shows that, in case q^O (mod 4), arbitrarily chosen β0 satisfies (i).
In case q = 0 (mod 4), put
2
(IΛTΓ) T-(lΛπ)-(S
i)β0=y(S2i)v(πΛπ),
y^Z2.
If y = 0, then /50 satisfies (i). If jΦO, put βΌ = βo + η(πΛπ), then /SJ
satisfies (4. 90 and (i) as is easily checked.
Thus the existence of β0 satisfying (4. 90 and (i) was proved.
The proof of the existence of α 0 satisfying (4.90 and (ii) is completely parallel to the above, and is left to the readers, q. e. d.
103
MULTIPLICATIVE STRUCTURES IN MOD q COHOMOLOGY THEORIES I.
From the above proof and (4. 9") we see that
(4.12) the elements a0 and β0 satisfying (4. 90 and the relations (i) and
(ii) of (4.11) is unique if q^2 (mod 4) // q=2 (mod 4), any elements
a0 and β0 satisfying (4. 90 satisfy (i) and (ii) of (4.11).
4.6. Next we compute the groups {Mqy Mt) (q,t>ϊ)y
we get the following exact sequence
2
, Mt}®Zq
By (1.7)
{Mqy
Let d=(q, t) be the greatest common divisoa of g and t. Tor ({S\ Mt}, Zq)
is isomorphic to Zd and generated by (t/d)-it. By (2.5).
From (4.2) it follows that {Mqy Mt} is generated by q/d and itηπqy
where d(q/d) = 0 or itvπqy and ityπq^0 if and only if q and t are even.
We have
(4.130 d(q/d) = itvπ^0, i. e., q/d is of order 2d, if and only if q~t = 2
(mod 4).
To see (4.130, we may assume that q and t are even, q/d or t/d
is odd since they are relatively prime. Assume that q/d is odd. Then,
using Theo. 1.1,
d-(q/d) = (q/d)d (q/d) = q.(q/d) = 0
if
(mod 4)
and, if q=2 (mod 4)
d (q/d) = q (q/d) = (q/d)iqyπq = (t/d)-itvπq
by (2. 5), which prove (4.130 in case q/d is odd. In case t/d is odd,
(4.130 can be proved similarly.
From (4.130
an
d the above exact sequence we obtain
(4.13).
q or / : odd
generators
where d = (q, t).
q = t=2 (mod 4)
zd
%2d
qjd
qjd
others
zd+z2
q/d, it7iπq
104
S. ARAKI and M. TODA
Let q>l, r^tly and consider the elements
(iqAig)v(S2πqr)^
2
2
{SMqry MqAMq}
and (S iqr)v(πqAπq)tΞ
{MqAMqy S Mgr}.
By (4.13),
2
(Sig)v(β πgr)Φ0
if and only if q and qr are even. The kernel of the
homomorphism
( l Λ i ) * : {SMqry SMq}->{SMqr,
MqΛMq)
is q{SMqr> SMq} which is generated by (Siq)v(Sπqr)
and vanishes otherwise. Thus we conclude that
(4.14) (igAig)v(S2πgr)=t=0
(mod 4).
if and only if
if q=qr=2
(mod 4)
q=0 (mod 4) or q=2, qr = 0
Similarly we see that
(4.140 (S2igr)η(πg/\πg)m0
(mod 4).
if and only if q = 0 (mod 4) or q=2,
qr=0
The following table (4.15) and the relation (4.150 are verified from
(1. 7), (1. 70, (4. 2) and Theo. 1.1 (c/., Theo. 4.1).
(4.15)
3
3
{S , Mq A Mg}^{Mg A Mq, S } ^
3
q : odd
g=0 (mod 4)
g=2 (mod 4)
zq
Zq + Z2
Z2q
generators of {S , Mq A Mg}
a0
generators of {Mq Λ Mg, S3}
βo
a0
βo, ηίπqΛ.πq)
where a0 and β0 are arbitrarily chosen elements satisfying
(4.150
q-ao = (iqAiq)v^O
|
and q βo = v(πqΛπq)^0
βo
(4. 90-
if q=2 (mod 4).
4.7. Lemma 4.6. (i) There exist sequences {aoq\, {βOtq}, q>l, of
3
elements aoq^{S\
Mq A Mq} and βoq^{MqAMqy
S } which satisfy (4.90,
(4.11) and
(4.16)
l:Mq^Mqr
r-βo>q for the maps r :
Mqr-+Mq,
of (2.5), r ^ l .
(ii) There
and (4.16); aOιg
(mod 4).
(iii) There
and (4.16); βo>q
(mod 4).
are just two sequences {α0 q} and {a^ q} satisfying (4. 90
= aίtg if q^2 (mod 4) and aΌ>q = a0>q + (iqAig)v
if q=2
are just two sequences {β0 q} and {β'Oj q} satisfying (4. 90
= βί>q if q^2 (mod 4) and β'»,q = β0>q + v(πqAπq) if q=2
MULTIPLICATIVE SSRUCTURES IN MOD q COHOMOLOGY THEORIES I.
(iv) // two sequences {a0 q} and {β0> q} satisfy
then they satisfy (4.11).
105
(4.90 and (4.16),
Proof, (i) For each q>l choose a pair of elements {ccoq} and {βΌ'fq}
satisfying (4. 90 and (4.11).
For each (q, r)9 q>l, r ^ l the element
(7/\τ)a'<! qr-r*a'^q
(r:Mqr->Mq)
is in the kernel of
- {S3, S*Mq}
( l Λ * , ) * : {S\ MqΛMq}
( l Λ ^ ) r α ί ; g = r S2ίff
since
/
(lAπq)(rAr)a o[gr
by (4.90
/ /
o >gr
by (2.5)
2
by (4.90
= (rAr'πgr)a
= r.{r/\l)S iqr
2
by (2.5).
= r-S iq
By (1.70 and (4.2), the kernel of (lΛ*r,)* is generated by
= (igΛiq)v which vanishes if and only if q is odd. Hence
(lΛig)(Sig)v
where xqr^Z2
if q is even, and xqr = 0 if q is odd.
Compose (FΛf) to the equation
of (4.11) from the left, where Tqr=T(Mqr,
Mqr).
(1. 2), (2. 5), (4.11) and (t| 1), we have that
Then, making use of
the left
side =
= (T q +1)(1 Λiq)Sf - αS;,(Sτrr)Sr =
the right side =
i. e.,
xqtr(iqΛig)v(Sπqr)
r*aζq(Sπqr),
=0.
Hence, by (4.14), we have
(l|2)
^ , = 0 if ^ Ξ O (mod 4) or if q=2f qr = 0 (mod 4 ) .
From (Ijl) and (t(2) we have
(1?3)
(rΛr)a'Ό'fgr
Now we put
= r aXg if q^2
(mod 4) or ^ r ^ 2 (mod 4 ) .
106
S. ARAKΪ and H. TODA
'.o,g
'Ό',q +χq/2,2'{igf\iq)
if
q^2 (mod 4)
if q=2 (mod 4 ) .
By (4 12) the sequence {α0 q} satisfies (4. 90 and (4.11). We shall
check (4.16) for this sequence {aoq}.
In case q^2 and qr^2 (mod 4),
(4.16) is obvious by (tf 3) and (ft 4). In case q^2 and qr=2 (mod 4), q
is odd, hence (iqAiq)v = 0 and (4.16) is easily seen. In case q=2 and
qr^2 (mod 4), r is even and we obtain (4.16) as follows:
(f Ar)a0>qr
= (?A?)af*\qr = r
aζ.
Finally consider the case q=qr=2 (mod 4). Since {aoq}
and (4.11), we can put, by (I) 1),
for some yqr^Z2.
Compose (q/2Aq/2), q]2:M2-*Mq,
from the left. Then, by (J|4), (2.7) and (2.5),
the left
to this equation
side =
(qr/2Λqr/2)(aί'tg+xgr/292(igrΛig,)v)
= (qr/2) -aΌ',2+2- xqr/2t 2 (i2 A i2) v
r'(q/2Aq/2)(a/0[q-Jtxq/2>2(iqAiq)v)+yg,r(i2Ai2)v
the right siάe =
r-((q/2)-aΌ/f2+2xq/2t2(i2Ai2)v)+yg,Λi2Λi2)η
=
i. e.,
satisfies (4.90
yg,r(i2Ai2)v
Since (i2Ai2)v^0
= 0.
by (4.150 we obtain
which proves (4.16) for the considered case.
Thus we have proved the existence of the sequence {α0> q} satisfying
(4. 90, (4.11) and (4.16).
The proof of the existence of a sequence {β0 q} satisfying (4.90,
(4.11) and (4.16) is completely dual to the above one, and the details
are left to the readers.
(ii) Assume that {aOt q} and {aίt q} satisfy (4.90 and (4.16). By
(4. 9") we can put
where zq = 0 if q is odd. Let q be even.
By (4. 16) and (2.5)
MULTIPLICATIVE STRUCTURES IN MOD q COHOMOLOGY THEORIES I.
Since (i2Ai2)v^0
by (4.150, we have
(q/2)z2 = zq
i. e.,
107
(mod 4),
zq = 0
if
^ΞO
(mod 4),
= z2
if
q=2 (mod 4 ) .
Thus there are at most two sequences satisfying (4. 9') and (4.16).
Next let {a0 q) be a sequence of (i). Put a'ϋ' q = aQq for q>2 and
«o%=α O i 2 + (f2Λi2)i7, then {<#,} satisfies (4.90 and (4.11) by (4.12).
Repeating the proof of (i) to this sequence {aΌ' q) we get a sequence
{a'otq} satisfying (4. 90 and (4.16). We see in (ij 4) that
since xlt2=0.
Thus {a0 q} and {aΌq} are two different sequences satisfying (4.90 and (4.16), and we have proved (ii).
(iii) The proof of (iii) is a dual of that of (ii).
(iv) By (ii), there are just two sequences {α0 q} and {a'Ό> q} satisfying (4. 90 and (4.16). But, as is seen in the last half of the proof of
(ii), both sequences are constructed by a method employed in the proof
of (i), hence they satisfy (4.11). Thus (iv) was proved, q. e. d.
4. 8. Take a pair of sequences {α0 q} and {βOt q} of Lemma 4. 6, (i).
In virtue of Lemma 4,4, (ii), we see that
(4.17) there exists a sequence {aq}f q>l, of elements cc = άq^ {Nq, Mq/\
Mq}> of which each element satisfies (4.8) and, putting όίqh = aoq = ao and
pPg = βo.g^βo*thesequences
{a0 q} and {/30>q} satisfy (4. 90, (4.11) and (4.16).
In the following, we fix a sequence {άq}
only the elements of this sequence.
We put
of (4.17), and use as a
(4.18)
.
a = aj<Ξ{Ng, MqΛMq}
This element a will play an important role in the following paragraphs. By (4. 8) and (4. 70 we have
(4.180
ai1 = aoy aio = iAi
and
(l
108
S. ARAKI and H. TODA
Proposition 4. 7. (i) (lMΛπ)Ta
(ϋ)
Proof,
= π0.
T(1*Λ i) + ( I M Λ i) = aix (Sπ).
(i)
(1 M Λ π) Ta = (1 M Λ π) a + (S 2 i) βoa
by (4. 9), (4.11), (4.18), (4.180 and (4. 8).
(ii) follows from (4.11), (ii) and (4.180-
q. e. d.
5. Existence of admissible multiplications
5.1. Let μ be an associative multiplication in a reduced cohomology
theory h. In this paragraph we define a multiplication μq in h{ Zq)
for each q>l, and prove that μq is admissible. Thereby, we need some
assumptions on μ and h in case q=2 (mod 4).
Using the notations of 4. 3, the cofibration
yields, for any object (finite C W-complex with a base point) I f of A, a
cofibration
WΛNq ^
g
In the exact sequence of Ji associated with this cofibration,
(lwΛSng)*
:ϊίk(WΛSn+2)-^Hk(WAMqASn+1)
becomes a trivial map if q^2 (mod 4), or if q=2 (mod 4) and (??7r)** = 0,
since the attaching map g*=0 in the former case and =η(Sπ) in the
latter case. Thus,
(5.1) the h-cohomology sequence associated with the above cofibration
breaks into the following short exact sequences
if qm2 (mod 4), or if q=2 (mod 4) and ( ηπ)** = 0 in h.
When q^2 (mod 4), Ng=S2VS2Mg
2
i':S Mq^Ng
and
by (4.6), (i). Let
2
π':Nq-*S
be the inclusion and the map collapsing S2Mg respectively.
we have
Obviously
MULTIPLICATIVE
STRUCTURES IN MOD q COHOMOLOGY THEORIES I.
π'i0 = 1, πoi' = 1, π'i' = 0
(5. 2).
and
109
n't, = 0 .
Lemma 5.1. // (i) #ΞJΞ2 (mod 4), or if (ii) q=2 (mod 4) and η** = 0
in h, then there exists an element γ 0 of h2{Nq) satisfying
2
(5.3)
/o*7o = σ l
and
In case (i), 7 0 =7r / H c (a 2 l) satisfies these relations.
If (iii) # = 2 (mod 4) <2w<i (ηπ)** = 0 m A, then there exists
satisfying
(5.30
yo^
ίί7o = σM.
Proof. Case (i) follows from (5.2) by putting <γo=π'*(σ2l).
In cases
(ii) and (iii), (^τr)**=O. Then, by (5.1) for Tf-S°, there exists
yf^h2{Nq)
2
such teat /^7ί = σ- l. Thus the case (iii) is proved by putting τ o = 7o.
In the remaining case (ii) (4.6), (ii) implies that i1(S2q) is homotopic to
iQη. Thus (S2^)Hί/f7o = 0. From the exactness of the sequence
h2(S2Mq) - — U
h\Sz) ±-^U
h\S3)
follows that (S 2 /)*^ = ίi*7o for some jt;eA 2 (S 2 MJ.
Put
Then we have
i *To = ί o'Ύo — ί *7Γ?ΛΓ = cr21
f fγ 0 = ί f γ{ - ίfarί # = (S2/)* Λ - (π0 i j * Λ:
-0
and
by (4. 70.
q. e. d.
5. 2. Making use of y 0 of Lemma 5.1, hence at least under the assumption of (ηπ)**=0 if q=2 (mod 4), we define a homomorphism
7 =
<γw:hk(WΛNq)-+ϊιk(WΛS2Mg)
by the formula
(5. 4)
yw(x)
k
for x^% {WANq).
= (l^Λίrof-'CΛ-Mσ-'αwrΛio
Since
= (lΛίo)*ΛΓ,
is in the kernel of (\wAi0)*.
By (5.1), (l^Λτr 0 )*
is monomorphic. Thus the map γ is a well-defined homomorphism.
110
S. ARAKI and
H. TODA
As is easily checked by (5.2), we have
7w = (1 wΛi'T
(5. 4').
if
<7 ΐ 2 (mod 4) and γ 0 = π'*(σ21).
Lemma 5.2. (i) ywisa left inverse of (I^A^Ό)*- i.e.,
= an identity map; hence the sequence of (5.1) splits:
(ii) 7 is natural in the sense that
where f .W'^W,
M=Mq
and
N=Ng.
(iii) γ is compatible with the suspension in the sense that
(VΛ T'Ύ*>γw = 7sπ(lwΛ T')*σ ,
where T' = T(S\ Nt)
(iv)
and T" = T(S\ S2Mg).
The relation
holds, where x^%k{WANg)
and y<=h>{Y).
Proof, (i) If x = (lΛπo)*y, then (lΛί o )** = 0 and (i) follows from
(5. 4).
(ii) Since (lw>Λπ0)* is monomorphic, it it sufficient to prove the
equality
= (±w>Aπo)*7w>(fAlN)*(x).
Now, the left ride = (fAπo)*7w(x) =
(fAlN)*(lwAπ0)*7w(x)
= the right side,
(iii) Since T o = T{S1, S2) is a map of degree 1, we have
(lswAio)*(lwΛ
T')*σ = (l^Λ T0)*<r
Making use of this identity, we have
2
T')*<rx-μ(<r- (hwAi0)*(lwA
T>)*σx®Ύo)
2
T')σx-(1WA
T')*σμ(σ- (lwAh)*x®7a)
MULTIPLICATIVE STRUCTURES IN MOD q COHOMOLOGY THEORIES I.
from which follows (iii) since (1sw/\π^f
(iv)
Ill
is monornorphic.
(lγΛW/\ τr0)* 7FΛ wμ(y® x)
from which follows (iv). q. e. d.
Lemma 5. 3. // a0 satisfies (5. 3), then the relation
holds for the inclusions i\Sl^Mq and il\
Proof.
Since S2i = πQil by (4.70 and f f γ 0 - 0 by (5.3), we see that
= (lw/\i)*x.
q. e. d.
To prove the quasi-associativity of the multiplication μq to be defined
in 5. 3, we need a special kind of commutativity, i. e.,
(5.5) if 70 = π'*(σ2l) in case q^2 (mod 4), or if μ is commutative, then
there holds a commutativity
(5.50
for any *efr'(Z), where
T'=T(Z,Nq\
The proof is clear,
Lemma 5. 4. // γ 0 satisfies (5. 50, then there holds a relation
(1WA T")*μ(7w(x)®z)
k
for any x^h (W/\Nq]
T"=T(Z,S2M,).
Proof.
and
= ΎWΛZ(IW/\
z^h*(Z\
T')*μ(x®z)
where
(l^ΛzΛτr 0 f(l^Λ T"J* μ(7w(x)®z)
Q* X® T'
Tf=T(ZyNq)
and
112
and
S. ARAKI and H. TODA
(l^ΛzΛ7r 0 )*γ W Λz(lwΛ T')* μ(x®z)
Thus (5. 50 concludes the lemma.
5. 3. Making use of the homomorphism γ defined by (5. 4) and the
element a<={Ng, Mq/\Mq] of (4.18), we define a map
(5. 6)
μq : h'(X; Zq)®fr(Y\
Zq)-+fo+'(X/\
Y;
as the composition
(5. 60
, =σ
\ Zq)®h'(Y\ Z,} = hi
μ
f\ γ/\M«)->hi+J+
Y/\Mq/\S2}
FΛM,) = fc+'(X/\ Y Zq) ,
where
T=T(Y,Mq).
μq is defined only if #ί2 (mod 4) or if q=2 (mod 4) and (ηπ}** = 0.
The definition of μg depends on the choices of γ 0 and a which are
but fixed during the subsequent proofs of properties of an admissible
multiplication.
Note that
(5.6") ^ = σ-W)**(lχΛ7V\l M )*
5. 4. Theorem 5. 5.
ing (ΛJ.
if
<?ΐ2 (mod 4) and γ0 = τr'*(σ2l) .
The map μq of (5. 6) is a multiplication satisfy-
Proof. The linearity and the naturality of μq is obvious.
To prove (ΛJ : putting
T'= T(F, Mq\ T,= T(S\ Y/\Mq\
T2
= T(S\ Mq) and T=T(Mq, Mq\ by definitions of Pq and μq we have
by Prop. 4.7, (i),
2 2
<τ~ σ μ = μ = μ^
by Lemma 5.2, (i).
MULTIPLICATIVE STRUCTURES IN MOD q COHOMOLOGY THEORIES I.
113
Similarly we see that
i. e., (ΛJ was proved.
From (Λj) and (ίf3) follows that pg(ΐ) is a bilateral unit of μgy i.e.,
the existence of lg is obtained.
To prove the compatibility of μq with σq : putting T=T(Y,Mq\
T,= T(Y,S*\ Tt=T(S\Mq\ T^T(S\Y/\Mq\
T,= T(S\ Yf\Nq\
T5
2
= T(S\Ng) and TB=T(S\S Mg)9 by definitions of μg and σq we have
Here
^ΛS 2 l M )*γ S X Λ r (l x Λ T4)*σ
r)*(lχΛΓ 4 )*σ
σ
y
by Lemma 5.2, (ii),
by Lemma 5.2, (iii).
Thus
T2)* σσ
Similarly we see that
.
q. e. d.
The above theorem, combined with Prop. 3. 3, shows
Corollary 5.6. When q~2 (mod 4), the condition that (τ77r)** = 0 in
h is necessary and sufficient for the existence of a multiplication μq
satisfying (ΛJ.
5. 5. Theorem 5. 7. // γ0 satisfies (5. 3), then the multiplication μq
of (5. 6) satisfies (Λ2).
Proof.
By Theo. 5.5
/
T=T(Mq, Mq\ T =T(Y,Mq)
we can use (ΛJ for μq. Putting
1
and T"- T(Y/\M9, S ), we have (on
114
S. ARAKI and H. TODA
; Z))
/
ΛlM)*A^
χΛ
.
by Prop. 4.7, (ii),
T'Λl M )V
by Lemma 5. 3,
q. e. d.
Theorem 5. 8. // γ0 satisfies
(5. 6) satisfies (Λ3).
(5. 5X), /Ae« /Aβ multiplication μq of
Proof. Since ^^ satisfies (ΛJ it is sufficient to prove the following
three relations :
(5.7)
(5. 70
μ*(μ*®ϊ) = μ«(l®μL) >
(5.7")
To prove (5.7): discussing on A*(Jf
putting T1=T(Z,Mg)9 we have
Z^)®^(F; Z,)<8)A*(Z; Zff) and
T^lj^* μ(μ®ΐ)
γΛT1ΛlM)*μ)
\T1/\lM)*μ)
by Lemma 5.2, (iv),
i. e., (5. 7) is proved.
X
In a similar way we can easily see (5. 7 ), and using Lemma 5. 4
instead of Lemma 5. 2, (iv), we can see (5. 7"). The details are left to
the readers, q. e. d.
5. 6. As a corollary of Theos. 5. 5, 5. 7, 5. 8, Lemma 5. 1 and (5. 5),
we obtain
Theorem 5.9, (Existence theorem). In case q ^2 (mod 4) admissible
MULTIPLICATIVE STRUCTURES IN MOD q COHOMOLOGY THEORIES I.
115
multiplications μq exist always In case q = 2 (mod 4), // we assume that
77^*^0 in h and μ is commutative, then admissible ones μq exist.
OSAKA CITY UNIVERSITY
KYOTO UNIVERSITY
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